推 math1209 :1.想想 LDT. 2. (1) 平均連續性 (2) 積分有限會怎樣? 05/05 19:32
1. Given a set E in [0,1] whith positive Lebesgue measure m(E)>0
and define f(x) = m(E∩[0,x]) for x in [0,1]. Prove that f is
differentiable a.e. and compute f' (a.e.) on E.
1
2. Let E be a Lebesgue measurable set on R with a finite Lebesgue
measure m(E) < ∞. For each x ≧ 0, we define f(x) = m(E∩E ),
x
where E = {x+y | y in E}. Prove that
x
(1) f is continuous on [0,∞).
(2) lim f(x) = 0.
x→∞
第一題要証f是幾乎處處可微，我是想f應該是單調增，所以就diff. a.e.
這樣對嗎?? 但後面要求f' 我就不知道該怎麼做了
第二題的話沒有想法...囧
請問這兩題要怎麼做呢?? 請指教 謝謝
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